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Odd cutsets and the hard-core model on ℤ d

Ron Peled, Wojciech Samotij (2014)

Annales de l'I.H.P. Probabilités et statistiques

We consider the hard-core lattice gas model on ℤ d and investigate its phase structure in high dimensions. We prove that when the intensity parameter exceeds C d - 1 / 3 ( log d ) 2 , the model exhibits multiple hard-core measures, thus improving the previous bound of C d - 1 / 4 ( log d ) 3 / 4 given by Galvin and Kahn. At the heart of our approach lies the study of a certain class of edge cutsets in ℤ d , the so-called odd cutsets, that appear naturally as the boundary between different phases in the hard-core model. We provide a refined combinatorial...

On a bifurcation problem arising in cholesteric liquid crystal theory

Carlo Greco (2017)

Commentationes Mathematicae Universitatis Carolinae

In a cholesteric liquid crystal the director field n ( x , y , z ) tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known as Freedericksz...

On a mathematical model for the crystallization of polymers

Maria Pia Gualdani (2003)

Bollettino dell'Unione Matematica Italiana

We consider a mathematical model proposed in [1] for the cristallization of polymers, describing the evolution of temperature, crystalline volume fraction, number and average size of crystals. The model includes a constraint W e ⁢ q on the crystal volume fraction. Essentially, the model is a system of both second order and first order evolutionary partial differential equations with nonlinear terms which are Lipschitz continuous, as in [1], or Hölder continuous, as in [3]. The main novelty here is the...

On a model of rotating superfluids

Sylvia Serfaty (2001)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an energy-functional describing rotating superfluids at a rotating velocity ω , and prove similar results as for the Ginzburg-Landau functional of superconductivity: mainly the existence of branches of solutions with vortices, the existence of a critical ω above which energy-minimizers have vortices, evaluations of the minimal energy as a function of ω , and the derivation of a limiting free-boundary problem.

On a model of rotating superfluids

Sylvia Serfaty (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an energy-functional describing rotating superfluids at a rotating velocity ω, and prove similar results as for the Ginzburg-Landau functional of superconductivity: mainly the existence of branches of solutions with vortices, the existence of a critical ω above which energy-minimizers have vortices, evaluations of the minimal energy as a function of ω, and the derivation of a limiting free-boundary problem.

On a nonlocal problem for a confined plasma in a Tokamak

Weilin Zou, Fengquan Li, Boqiang Lv (2013)

Applications of Mathematics

The paper deals with a nonlocal problem related to the equilibrium of a confined plasma in a Tokamak machine. This problem involves terms u * ' ( | u > u ( x ) | ) and | u > u ( x ) | , which are neither local, nor continuous, nor monotone. By using the Galerkin approximate method and establishing some properties of the decreasing rearrangement, we prove the existence of solutions to such problem.

On a variant of Korn’s inequality arising in statistical mechanics

L. Desvillettes, Cédric Villani (2002)

ESAIM: Control, Optimisation and Calculus of Variations

We state and prove a Korn-like inequality for a vector field in a bounded open set of ℝ N , satisfying a tangency boundary condition. This inequality, which is crucial in our study of the trend towards equilibrium for dilute gases, holds true if and only if the domain is not axisymmetric. We give quantitative, explicit estimates on how the departure from axisymmetry affects the constants; a Monge–Kantorovich minimization problem naturally arises in this process. Variants in the axisymmetric case are...

On a variant of Korn's inequality arising in statistical mechanics

L. Desvillettes, Cédric Villani (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We state and prove a Korn-like inequality for a vector field in a bounded open set of ℝ N , satisfying a tangency boundary condition. This inequality, which is crucial in our study of the trend towards equilibrium for dilute gases, holds true if and only if the domain is not axisymmetric. We give quantitative, explicit estimates on how the departure from axisymmetry affects the constants; a Monge–Kantorovich minimization problem naturally arises in this process. Variants in the axisymmetric case...

On bilinear kinetic equations. Between micro and macro descriptions of biological populations

Mirosław Lachowicz (2003)

Banach Center Publications

In this paper a general class of Boltzmann-like bilinear integro-differential systems of equations (GKM, Generalized Kinetic Models) is considered. It is shown that their solutions can be approximated by the solutions of appropriate systems describing the dynamics of individuals undergoing stochastic interactions (at the "microscopic level"). The rate of approximation can be controlled. On the other hand the GKM result in various models known in biomathematics (at the "macroscopic level") including...

On Cauchy problem for the equations of reactor kinetics.

Jan Kyncl (1989)

Aplikace matematiky

In this paper, the initial value problem for the equations of reactor kinetics is solved and the temperature feedback is taken into account. The space where the problem is solved is chosen in such a way that it may correspond best of all to the mathematical properties of the cross-section models. The local solution is found by the method of iterations, its uniqueness is proved and it is shown also that existence of global solution is ensured in the most cases. Finally, the problem of mild solution...

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